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FINITIST AXIOMATIC TRUTH
- Source :
- Sato, Kentaro; Walker, Jan (2023). Finitist Axiomatic Truth. The journal of symbolic logic, 88(1), pp. 22-73. Cambridge University Press 10.1017/jsl.2022.65
- Publication Year :
- 2022
- Publisher :
- Cambridge University Press (CUP), 2022.
-
Abstract
- Following the finitist’s rejection of the complete totality of the natural numbers, a finitist language allows only propositional connectives and bounded quantifiers in the formula-construction but not unbounded quantifiers. This is opposed to the currently standard framework, a first-order language. We conduct axiomatic studies on the notion of truth in the framework of finitist arithmetic in which at least smash function $\#$ is available. We propose finitist variants of Tarski ramified truth theories up to rank $\omega $ , of Kripke–Feferman truth theory and of Friedman–Sheard truth theory, and show that all of these have the same strength as the finitist arithmetic of one higher level along Grzegorczyk hierarchy. On the other hand, we also show that adding Burgess-style groundedness schema, adjusted to the finitist setting, makes Kripke–Feferman truth theory as strong as primitive recursive arithmetic. Meanwhile, we obtain some basic results on finitist theories of (full and hat) inductive definitions and on the second order axiom of hat inductive definitions for positive operators.
- Subjects :
- Philosophy
510 Mathematics
Logic
000 Computer science, knowledge & systems
Subjects
Details
- ISSN :
- 19435886 and 00224812
- Volume :
- 88
- Database :
- OpenAIRE
- Journal :
- The Journal of Symbolic Logic
- Accession number :
- edsair.doi.dedup.....f74e3256c9cb539d87d701c354539126